Jacobi transformation conjecture for VOA characters

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Let VV be a rational vertex operator algebra of CFT type, let MM be an irreducible VV-module, and let h{\mathfrak h} and L∘L^{\circ} be the spaces defined in the source, with L∘L^{\circ} the dual lattice of the lattice LL. For v∈hv\in{\mathfrak h} and α∈L∘\alpha\in L^{\circ}, write χM(v,τ)\chi_M(v,\tau) for the character associated with MM. Jacobi transformation conjecture.

χM(v+ατ,τ)=e−2πi(v,α)−πi(α,α)τχM(v,τ).\chi_M(v+\alpha\tau,\tau)=e^{-2\pi i(v,\alpha)-\pi i(\alpha,\alpha)\tau}\chi_M(v,\tau).

This predicts the elliptic transformation law for the characters of irreducible modules and is part of the expected Jacobi-form behavior of VOA characters. The source gives no further information about its resolution.

References

Primary source

Chongying Dong, Kefeng Liu and Xiaonan Ma, “Elliptic genus and vertex operator algebras”, arXiv:math/0201135 (2002).

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